I have about 11 years experience with German car ownership both MBs and BMWs. When you say the brakes are working normally and the ABS light is on, I don't think you have tested your brakes properly. The ABS does nothing during normal braking. It only does its thing during an emergency braking situation or braking hard in the wet.
The advice offered in the posts above is all valid. However, a small test can be done to know if your ABS is working or not. On a quiet street with no traffic, get your car to about 60km/h and slam as hard as you can on the brakes with all your strength. If the wheels lock, the ABS is not working. If you feel a pulsation in the brake pedal and the wheels don't lock up, ABS is working.
If the ABS is not working it is 90% that one of the four wheel sensor wire is broken. It is very rare that the modulator fails. My experience has taught me that the break in the wire is usually in the section of the wire between the grip (near the shock) to the sensor. It may be possible (though rare) that the sensor itself has failed.
BTW, does the ABS light double as a warning for something else? Brake pads? low brake fluid? Usually the ABS light is yellow while the other two are red. Do you have anti skid in your car? But usually anti skid (ASC or DSC) have a separate warning light.
